Regular ultrafilters and finite square principles. (English) Zbl 1160.03026

Kennedy, Juliette; Shelah, Saharon; Väänänen, Jouko

J. Symb. Log. 73, No. 3, 817-823 (2008).

Summary: We show that many singular cardinals \(\lambda \) above a strongly compact cardinal have regular ultrafilters \(D\) that violate the finite square principle \(\square ^{\text{fin}}_{\lambda ,D}\) introduced in [J. Kennedy and S. Shelah, J. Symb. Log. 67, No. 3, 1169–1177 (2002; Zbl 1012.03048)]. For such ultrafilters \(D\) and cardinals \(\lambda \) there are models of size \(\lambda \) for which \(M^{\lambda }/D\) is not \(\lambda ^{++}\)-universal and elementarily equivalent models \(M\) and \(N\) of size \(\lambda \) for which \(M^{\lambda }/D\) and \(N^{\lambda }/D\) are non-isomorphic. The question of the existence of such ultrafilters and models was raised in [C. C. Chang and H. J. Keisler, Model theory. 3rd rev. edition. Amsterdam: North-Holland (1990; Zbl 0697.03022)].

MSC:

03E05	Other combinatorial set theory
03C55	Set-theoretic model theory

Keywords:

singular cardinals; regular ultrafilters; finite square principle

Citations:

Zbl 1012.03048; Zbl 0697.03022

References:

[1]	DOI: 10.1007/s001530050052 · Zbl 0874.03060
[2]	More on regular reduced products 69 pp 1261– (2004)
[3]	DOI: 10.1007/BF02764857 · Zbl 0681.03016
[4]	DOI: 10.1016/0003-4843(72)90001-0 · Zbl 0257.02035
[5]	Model theory (1990)
[6]	On regular reduced products 67 pp 1169– (2002)